JEE Mains Previous Year Questions (PYQ)

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JEE Main 2024 9th April Morning Shift

The parabola $ y^2 = 4x $ divides the area of the circle $ x^2 + y^2 = 5 $ in two parts. The area of the smaller part is equal to:

6.

Let the latus rectum of the hyperbola $ \dfrac{x^2}{9} - \dfrac{y^2}{b^2} = 1 $ subtend an angle of $ \dfrac{\pi}{3} $ at the centre of the hyperbola. If $ b^2 $ is equal to $ \dfrac{l}{m}(1 + \sqrt{n}) $, where $ l $ and $ m $ are co-prime numbers, then $ l^2 + m^2 + n^2 $ is equal to $ \_\_\_\_\_\_\_\_ $.

7.

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

8.

Let the line $ L:\ \sqrt{2}x+y=\alpha $ pass through the point of the intersection $ P $ (in the first quadrant) of the circle $ x^2+y^2=3 $ and the parabola $ x^2=2y $. Let the line $ L $ touch two circles $ C_1 $ and $ C_2 $ of equal radius $ 2\sqrt{3} $. If the centres $ Q_1 $ and $ Q_2 $ of the circles $ C_1 $ and $ C_2 $ lie on the y-axis, then the square of the area of the triangle $ PQ_1Q_2 $ is equal to ______.

9.

For $ 0<\theta<\frac{\pi}{2} $, if the eccentricity of the hyperbola $ x^2-y^2cosec^2\theta=5 $ is $ \sqrt{7} $ times eccentricity of the ellipse $ x^2cosec^2\theta+y^2=5 $, then the value of $ \theta $ is:

10.

Let the line $ 2x+3y-k=0,\,k>0 $, intersect the $ x $-axis and $ y $-axis at the points $ A $ and $ B $, respectively. If the equation of the circle having the line segment $ AB $ as a diameter is $ x^{2}+y^{2}-3x-2y=0 $ and the length of the latus rectum of the ellipse $ x^{2}+9y^{2}=k^{2} $ is $ \frac{m}{n} $, where $ m $ and $ n $ are coprime, then $ 2m+n $ is equal to:

11.

Let the length of the focal chord $ PQ $ of the parabola $ y^2 = 12x $ be $ 15 $ units. If the distance of $ PQ $ from the origin is $ p $, then $ 10p^2 $ is equal to $ \_\_\_\_\_\_\_\_\_\_ $.

12.

Let $ L_1, L_2 $ be the lines passing through the point $ P(0,1) $ and touching the parabola $ 9x^2 + 12x + 18y - 14 = 0 $. Let $ Q $ and $ R $ be the points on the lines $ L_1 $ and $ L_2 $ such that the $ \triangle PQR $ is an isosceles triangle with base $ QR $. If the slopes of the lines $ QR $ are $ m_1 $ and $ m_2 $, then $ 16\left(m_1^2 + m_2^2\right) $ is equal to ____.

13.

Let $ PQ $ be a chord of the parabola $ y^2 = 12x $ and the midpoint of $ PQ $ be at $ (4, 1) $. Then, which of the following point lies on the line passing through the points $ P $ and $ Q $?

14.

Consider the circle $ C: x^2 + y^2 = 4 $ and the parabola $ P: y^2 = 8x $. If the set of all values of $ \alpha $, for which three chords of the circle $ C $ on three distinct lines passing through the point $ (\alpha, 0) $ are bisected by the parabola $ P $ is the interval $ (p, q) $, then $ (2q - p)^2 $ is equal to ________.

15.

The length of the latus rectum and directrices of a hyperbola with eccentricity $ e $ are $ 9 $ and $ x = \pm \frac{4}{\sqrt{3}} $, respectively. Let the line $ y-\sqrt{3}x+\sqrt{3}=0 $ touch this hyperbola at $ (x_0,y_0) $. If $ m $ is the product of the focal distances of the point $ (x_0,y_0) $, then $ 4e^2+m $ is equal to ______.

16.

Let $ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $, $ a>b $ be an ellipse, whose eccentricity is $ \frac{1}{\sqrt{2}} $ and the length of the latusrectum is $ \sqrt{14} $. Then the square of the eccentricity of $ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $ is:

17.

The length of the chord of the ellipse $ \dfrac{x^2}{25} + \dfrac{y^2}{16} = 1 $, whose mid-point is $ \left( 1, \dfrac{2}{5} \right) $, is equal to:

18.

Let $ A $, $ B $ and $ C $ be three points on the parabola $ y^2 = 6x $ and let the line segment $ AB $ meet the line $ L $ through $ C $ parallel to the $ x $-axis at the point $ D $. Let $ M $ and $ N $ respectively be the feet of perpendiculars from $ A $ and $ B $ on $ L $. Then $ \left(\frac{AM \cdot BN}{CD}\right)^2 $ is equal to ________.

19.

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$ and eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to:

20.

Let $ f(x)=x^{2}+9 $, $ g(x)=\dfrac{x}{x-9} $ and $ a=f o g(10),\, b=g o f(3) $. If $ e $ and $ l $ denote the eccentricity and the length of the latus rectum of the ellipse $ \dfrac{x^{2}}{a}+\dfrac{y^{2}}{b}=1 $, then $ 8e^{2}+l^{2} $ is equal to:

21.

Let $ P $ be a point on the ellipse $\displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1 $. Let the line passing through $ P $ and parallel to $ y $-axis meet the circle $ x^2 + y^2 = 9 $ at point $ Q $ such that $ P $ and $ Q $ are on the same side of the $ x $-axis. Then, the eccentricity of the locus of the point $ R $ on $ PQ $ such that $ PR:RQ = 4:3 $ as $ P $ moves on the ellipse, is:

22.

Let $ S $ be the focus of the hyperbola $ \dfrac{x^{2}}{3}-\dfrac{y^{2}}{5}=1 $, on the positive $ x $-axis. Let $ C $ be the circle with its centre at $ A(\sqrt{6},\sqrt{5}) $ and passing through the point $ S $. If $ O $ is the origin and $ SAB $ is a diameter of $ C $, then the square of the area of the triangle $ OSB $ is equal to ________.

23.

Let $ A(\alpha, 0) $ and $ B(0, \beta) $ be the points on the line. Let the point $ P $ divide the line segment $ AB $ internally in the ratio $ 7:3 $. Let $ 3x - 25 = 0 $ be a directrix of the ellipse $ E: \, \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $ and the corresponding focus be $ S $. If from $ S $, the perpendicular on the $ x $-axis passes through $ P $, then the length of the latus rectum of $ E $ is equal to:

24.

If the shortest distance of the parabola $ y^2 = 4x $ from the centre of the circle $ x^2 + y^2 - 4x - 16y + 64 = 0 $ is $ d $, then $ d^2 $ is equal to:

25.

Suppose $ AB $ is a focal chord of the parabola $ y^{2} = 12x $ of length $ l $ and slope $ m < \sqrt{3} $. If the distance of the chord $ AB $ from the origin is $ d $, then $ l d^{2} $ is equal to __________.

26.

Let $ H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1 $ be the hyperbola, whose eccentricity is $ \sqrt{3} $ and the length of the latus rectum is $ 4\sqrt{3} $. Suppose the point $ (\alpha, 6) $, $ \alpha > 0 $ lies on $ H $. If $ \beta $ is the product of the focal distances of the point $ (\alpha, 6) $, then $ \alpha^2 + \beta $ is equal to:

27.

Let the foci of a hyperbola $ H $ coincide with the foci of the ellipse $\displaystyle E:\ \frac{(x-1)^{2}}{100}+\frac{(y-1)^{2}}{75}=1 $ and the eccentricity of the hyperbola $ H $ be the reciprocal of the eccentricity of the ellipse $ E $. If the length of the transverse axis of $ H $ is $ \alpha $ and the length of its conjugate axis is $ \beta $, then $ 3\alpha^{2}+2\beta^{2} $ is equal to

28.

If the points of intersection of two distinct conics $ x^2 + y^2 = 4b $ and $ \dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1 $ lie on the curve $ y^2 = 3x^2 $, then $ 3\sqrt{3} $ times the area of the rectangle formed by intersection points is __________.

29.

Let $ e_1 $ be the eccentricity of the hyperbola $\displaystyle \frac{x^2}{16}-\frac{y^2}{9}=1 $ and $ e_2 $ be the eccentricity of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b $, which passes through the foci of the hyperbola. If $ e_1 e_2=1 $, then the length of the chord of the ellipse parallel to the $ x $-axis and passing through $ (0,2) $ is:

30.

Let the foci and length of the latus rectum of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \ a > b $ be $ (\pm 5, 0) $ and $ \sqrt{50} $, respectively. Then, the square of the eccentricity of the hyperbola $ \frac{x^2}{b^2} - \frac{y^2}{a^2 b^2} = 1 $ equals:

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